Why are there 2 pi radians in a circle?
One definition says the radian measure of angle
The arc length of the whole circle is its circumference
This definition is the reason that we have the formula for arc length
Note that whatever unit are used to measure
Also note that some teachers introduce radian measure without discussing the 'official' definition as a ratio of lengths. They simply announce that once around the circle is
This makes it look arbitrary when it is not arbitrary at all.